4.3.1 Goal
The goal of this test is to validate the energy limit (Emax) set for a safe human handling of the
Panda robotic arm as a haptic input device. 4.3.2 Method
To validate the energy limit, the total energy of the system (eq. 3.1 in Chapter 3) defined as the sum of the kinetic energy of the master device, and the potential energy due to the spatial spring should either be lower or equal withEmaxor else, a factor term (λ) is used to reduce the
computed wrench (Wp) applied to the user by the Panda robotic arm.
Similar to the first test, for initialization (P0) two safety limits are set: 198 W for the maximum power allowed between the master and user; and 140 N for the force allowed between the mas- ter and user. And unlike the first test, for the safety limit herein tested, 3.5 J is set as the max- imum total energy that may be allowed between the master and slave devices, in order to re- duce the effort required by the user to reach the limit during the experiment.
While viewing on a screen the slave device simulated in Gazebo environment where there is also a wall, the user moves the master end-effector slowly such that the slave end-effector can be seen approaching the virtual wall on the screen during free motion (P1). When the slave end-effector reaches the wall, the user continues to move the master end-effector (P2) before moving it back to its initial position and approaching the wall location again (P3). The user repeats two more times the interaction with the wall for reproducibility (P4, P5 and P6). 4.3.3 Expected results
At the beginning (P0) of the experiment, both the master and slave end-effectors have identical starting configurations and are not in motion. Therefore, the total energy of the system (see equation 3.1) is expected to be zero due to the potential energy (see equation 3.3) and kinetic energy (see equation 3.2) being zero. During the interaction with the wall (P2, P4 and P6), the total energy is expected to exceed theEmaxvalue and, as a result, the wrench feedback should
be limited accordingly for the same time periods (P2, P4, P6). 4.3.4 Results and discussion
The total energy of the system is found to be not zero, but−75 J (Figure 4.7a), given the control- ler parameters chosen (Kt=500I;Ko=50I;Kc=0I). Therefore, the output force and torque
are not restricted, and user safety is not guaranteed as such. In detail, the total energy value is calculated to be−75 J due to the orientation component of the total energy potential (equation 3.4). Assuming identical configurations (i.e.Rssp=I;p
sp
s =0) and eq. 3.4, eq. 4.3 holds. Vt(Rssp,p sp s )=0 Vo(Rssp)= −tr ¡ GoRsps ¢ = −75 Vc(Rssp,p sp s )=0 (4.3)
where tr() is the tensor trace operator that computes the sum of the diagonal elements;Rsspis
the rotation matrix from the frame orientation of the master end-effector to the one of the slave end-effector;psps is the position vector between the location of the master end-effector to the
one of the slave end-effector; andGois the rotational co-stiffness matrix computed from the
rotational stiffness matrix (Ko) by eq. 4.4. Go=
1
2tr(Ko)I−Ko=25I (4.4) Therefore, the value is to be considered as an offset determined by the way the rotational po- tential energy is defined in the original formula itself (Tadele, 2014; Raiola et al., 2018). As it is,
the implemented energy safety limit can not be validated. The computed total energy is signi- ficantly higher than−75 J during the interaction with the virtual wall (P2, P4 and P6) but values are always too low to ever trigger the energy safety limit (Emax), which is set to 3.5 J. The exact
cause of the deviation was not explored in this study due to time restrictions.
Figure 4.7:The system behaviour using the original formula(eq. 3.4) for computing the potential energy of the spatial spring. The total energy system (a); the magnitude force (b) and torque (c) output of the master device. The free motion parts are illustrated (P1, P3 and P5) within the figure, while the interaction with the wall (P2, P4 and P6) are not label within the figure.
A simple way to deal with the offset could be including it in the rotational potential energy (Vo(Rsps )) computation by adding a constant value computed from the controller parameter
chosen (equation 4.5). Vo(Rssp)= −tr ¡ GoRssp ¢ +tr(Go) (4.5)
Note that, all signals monitored during the wall interaction P2, P4 and P6 (i.e. the total energy, stiffness factor, force and torque as computed from the impedance controller) have a noisy peak as shown in Appendix B (see figures B.2 and B.1). The noisy instances occur when the stiffness parameters of the impedance controller are constantly updated to reduce the output force and torque. To clean the signals a 5-term moving average filter was applied.
The filtered results give a total energy as depicted in Figure 4.8. Both the force and the torque are limited during the wall interaction once the total maximum energy (Emax) is exceeded (see
Table 4.5). When the computed total energy exceeds theEmaxset, the force magnitude does not
exceed 58 N (Figure 4.9), which is also the mean value approximation over all wall interaction parts (P2, P4, P6) considering the respective standard deviations. Similarly for the torque, when the total energy exceeds theEmaxset, the torque magnitude does not exceed 55 Nm, the mean
value approximation over all wall interaction parts (P2, P4, P6) is 51 Nm. The consistency of all mean values being close to the maximum values and of all standard deviation values being low proves the efficiency of the energy layer in restricting the output force and torque in situations that are unsafe for human users (Etot≥Emax).
Force (N) Torque (Nm) Part Duration while
Etot≥Emax(s) Max Mean Std. Dev. Max Mean Std. Dev.
P2 1.15 57.86 56.71 1.09 54.21 51.86 1.74 P4 2.47 57.35 56.69 0.53 51.42 50.49 0.82 P6 3.53 57.88 57.01 0.68 51.22 49.94 1.00
Table 4.5:The maximum, mean and standard deviation of the force and torque while the total energy (Etot) exceeds the total maximum energy (Emax) due to the interaction with the wall (P2, P4 and P6), computed from the filtered data (Figure 4.9).
Herein, the first set of results obtained by following the energy potential definition reported by Fasse (1997) and Stramigioli (2001), and used by Raiola et al. (2018), do not share the offset seen in the rotational potential energy computation. It is unclear how Raiola et al. (2018) did not encounter the same issue and if the tested formula (eq. 4.5) is the proper way to deal with it. Moreover, this requires further investigation which is out of the scope of this project. Non- etheless, the new approach used here, that includes the offset in the rotational potential energy (Vo(Rssp)), yielded the expected results.
Figure 4.8:The total energy of the system (a) and the stiffness factorλ(b), smoothen with a 5-term moving average filter. The free motion parts are illustrated (P1, P3 and P5) within the figure, while the interaction with the wall (P2, P4 and P6) are not label within the figure.
Figure 4.9:The magnitude of the output force (a) and torque (b) of the master device, smoothen with a 5-term moving average filter. The free mo- tion parts are illustrated (P1, P3 and P5) within the figure, while the interaction with the wall (P2, P4 and P6) are not label within the figure.